Section author: Danielle J. Navarro and David R. Foxcroft

The continuity correction

Okay, time for a little bit of a digression. I’ve been lying to you a little bit so far. There’s a tiny change that you need to make to your calculations whenever you only have 1 degree of freedom. It’s called the “continuity correction”, or sometimes the Yates correction. Remember what I pointed out earlier: the χ² test is based on an approximation, specifically on the assumption that the binomial distribution starts to look like a normal distribution for large N. One problem with this is that it often doesn’t quite work, especially when you’ve only got 1 degree of freedom (e.g., when you’re doing a test of independence on a 2 × 2 contingency table). The main reason for this is that the true sampling distribution for the χ²-statistic is actually discrete (because you’re dealing with categorical data!) but the χ² distribution is continuous. This can introduce systematic problems. Specifically, when N is small and when df = 1, the goodness-of-fit statistic tends to be “too big”, meaning that you actually have a bigger α value than you think (or, equivalently, the p-values are a bit too small).

Yates (1934) suggested a simple fix, in which you redefine the goodness-of-fit statistic as:

\[\chi^2 = \sum_{i} \frac{(|E_i - O_i| - 0.5)^2}{E_i}\]

Basically, he just subtracts off 0.5 everywhere.

As far as I can tell from reading Yates’ paper, the correction is basically a hack. It’s not derived from any principled theory. Rather, it’s based on an examination of the behaviour of the test, and observing that the corrected version seems to work better. You can specify this correction in jamovi from a check box in the Statistics options, where it is called χ² continuity correction.